Chapter 12 - Vectors and the Geometry of Space - 12.4 Exercises - Page 839: 47

$|a \times b|^2=|a|^2 |b|^2-(a \cdot b)^2$

$|a \times b|^2=( |a| |b||sin \theta|)^2= |a|^2 |b|^2sin ^2\theta$ As we know : $sin ^2 \theta =1 -cos ^2 \theta$ Thus, $ |a|^2 |b|^2sin ^2\theta= |a|^2 |b|^2-|a|^2 |b|^2cos ^2\theta$ Remember that $(a \cdot b)^2=(|a|^2 |b|^2cos\theta) ^2=|a|^2 |b|^2cos^2\theta$ Hence, $|a \times b|^2=|a|^2 |b|^2-(a \cdot b)^2$